Metamath Proof Explorer
Description: A weak universe is closed under subsets. (Contributed by Mario
Carneiro, 2-Jan-2017)
|
|
Ref |
Expression |
|
Hypotheses |
wununi.1 |
⊢ ( 𝜑 → 𝑈 ∈ WUni ) |
|
|
wununi.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑈 ) |
|
|
wunss.3 |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 ) |
|
Assertion |
wunss |
⊢ ( 𝜑 → 𝐵 ∈ 𝑈 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
wununi.1 |
⊢ ( 𝜑 → 𝑈 ∈ WUni ) |
2 |
|
wununi.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑈 ) |
3 |
|
wunss.3 |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 ) |
4 |
1 2
|
wunpw |
⊢ ( 𝜑 → 𝒫 𝐴 ∈ 𝑈 ) |
5 |
1 4
|
wunelss |
⊢ ( 𝜑 → 𝒫 𝐴 ⊆ 𝑈 ) |
6 |
2 3
|
sselpwd |
⊢ ( 𝜑 → 𝐵 ∈ 𝒫 𝐴 ) |
7 |
5 6
|
sseldd |
⊢ ( 𝜑 → 𝐵 ∈ 𝑈 ) |